No abstract

We derive the partial integro-differential equations (PIDEs) verified by the values of European and barrier options in exponential Lévy models. We discuss the conditions under which options prices are classical solutions of the PIDEs. Since these conditions may fail in general, we consider the notion of continuous viscosity solution. We give sufficient conditions on the Lévy triplet for the option price to be continuous; in this case we show that it is the unique viscosity solution of the PIDE.

We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Lévy process or, more generally, a time-inhomogeneous jumpdiffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Lévy measure. We propose an explicit-implicit time-stepping scheme to solve the equation and study stability and convergence of the schemes proposed, using the notion of viscosity solution. Numerical tests are performed with smooth and non-smooth initial conditions. Our scheme can be used for European and barrier options, applies in the case of pure-jump models or degenerate diffusion coefficients, and extends to time-dependent coefficients.

Abel Symposium 2005 on Stochastic analysis and applications in honor of Kiyosi Ito's 90th birthday. AbstractWe consider the problem of hedging a contingent claim, in a market where prices of traded assets can undergo jumps, by trading in the underlying asset and a set of traded options. We give a general expression for the hedging strategy which minimizes the variance of the hedging error, in terms of integral representations of the options involved. This formula is then applied to compute hedge ratios for common options in various models with jumps, leading to easily computable expressions. The performance of these hedging strategies is assessed through numerical experiments.

Most authors who studied the problem of hedging an option in incomplete markets, and, in particular, in models with jumps, focused on finding the strategies that minimize the residual hedging error. However, the resulting strategies are usually unrealistic because they require a continuously rebalanced portfolio, which is impossible in practice due to transaction costs. In reality, the portfolios are rebalanced discretely, which leads to a 'hedging error of the second type', due to the difference between the optimal strategy and its discretely rebalanced version. In this paper, we analyze this second hedging error and establish a limit theorem for the renormalized error, when the discretization step tends to zero, in the framework of general Itô processes with jumps. Theses results are applied to hedging options with discontinuous payoffs in jump-diffusion models.

No abstract

Option pricing in models with jumps leads to the solution of partial integro‐differential equations (PIDEs). We present the specific features of the PIDEs compared with the partial differential equations. In particular, we highlight the difficulties that arise for the numerical solution of these equations. We show that the use of standard methods, such as finite differences or finite elements, is not straightforward. We then present a simple explicit–implicit finite difference scheme for pricing European vanilla and barrier options in exponential Lévy models and survey the other existing numerical techniques. We also discuss American option pricing, which leads to variational inequalities with the same integro‐differential operator.